This invention is a method and apparatus for heat exchange. Its main applications are in distillation and in counterflow heat exchange. Various models of the invention distill water, distill fuel alcohol, concentrate juices and brines, separate toxic chemicals from industrial wastewater, remove moisture from grains or clothing, exchange heat between liquids, and generate electrical power. The fundamental design strategy of the invention is to move heat over a large surface area relative to the flow rate through the system. This background section identifies some of the problems addressed by the invention and describes the conventional solutions.